Determine the net change and average rate of change between the values of the variable

Question

A function is given.

f(x) = 3 - 3x^2; x = 4, x = 4 + h

determine the net change between the given values of the variable.

determine the average rate of change between the given values of the variable.

Answer 1

Let's define these terms first:

Net change: f(x+h) - f(x)

Average rate of change: [f(x+h) - f(x)] / [(x + h) - x]

Solving for each we get the following:

Net change:

f(x+h) - f(x) = [3 - 3*(x + h)^2] - [3 - 3*x^2]

= [3 - 3*x^2 - 6*x*h - 3*h^2] - [3 - 3*x^2]

= -6*x*h - 3*h^2

= -3*h(2*x +h)

Average rate of change:

[f(x+h) - f(x)] / [(x + h) - x] = -3*h(2*x + h) / [(x+h) - x]

= -3*h(2*x + h) / h

= -3(2*x + h)

= -6x - 3h